Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation |x|=-1
-
Equation tan(x)=7
-
Equation 6-3*(x+1)=7-2*x
-
Equation x^6+1=0
- Express {x} in terms of y where:
- 13*x-1*y=-4
- -17*x+6*y=14
- -15*x-5*y=12
- -5*x+7*y=3
- Graphing y =:
- x^6+1
- Factor polynomial:
- x^6+1
-
Identical expressions
- x^ six + one = zero
- x to the power of 6 plus 1 equally 0
- x to the power of six plus one equally zero
- x6+1=0
- x⁶+1=0
- x^6+1=O
-
Similar expressions
- x^6-1=0
x^6+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{6} + 1 = 0$$
Because equation degree is equal to = 6 and the free term = -1 < 0,
so the real solutions of the equation d'not exist
All other 6 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = -1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{6 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = -1$$
so
$$\cos{\left(6 p \right)} = -1$$
and
$$\sin{\left(6 p \right)} = 0$$
then
$$p = \frac{\pi N}{3} + \frac{\pi}{6}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - i$$
$$z_{2} = i$$
$$z_{3} = - \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$z_{4} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$z_{5} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$z_{6} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - i$$
$$x_{2} = i$$
$$x_{3} = - \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$x_{4} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$x_{5} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$x_{6} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$x^{6} + 1 = 0$$
Because equation degree is equal to = 6 and the free term = -1 < 0,
so the real solutions of the equation d'not exist
All other 6 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = -1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{6 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = -1$$
so
$$\cos{\left(6 p \right)} = -1$$
and
$$\sin{\left(6 p \right)} = 0$$
then
$$p = \frac{\pi N}{3} + \frac{\pi}{6}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - i$$
$$z_{2} = i$$
$$z_{3} = - \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$z_{4} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$z_{5} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$z_{6} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - i$$
$$x_{2} = i$$
$$x_{3} = - \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$x_{4} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$x_{5} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
$$x_{6} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
Sum and product of roots
[src]
sum
___ ___ ___ ___
I \/ 3 I \/ 3 \/ 3 I I \/ 3
-I + I + - - - ----- + - - ----- + ----- - - + - + -----
2 2 2 2 2 2 2 2
$$\left(\left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right) + \left(\left(\left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right) + \left(- i + i\right)\right) + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)\right)\right) + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$$
=
0
$$0$$
product
/ ___\ / ___\ / ___ \ / ___\
| I \/ 3 | |I \/ 3 | |\/ 3 I| |I \/ 3 |
-I*I*|- - - -----|*|- - -----|*|----- - -|*|- + -----|
\ 2 2 / \2 2 / \ 2 2/ \2 2 /
$$- i i \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right) \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right) \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right) \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$$
=
1
$$1$$
1
Rapid solution
[src]
x1 = -I
$$x_{1} = - i$$
x2 = I
$$x_{2} = i$$
___
I \/ 3
x3 = - - - -----
2 2
$$x_{3} = - \frac{\sqrt{3}}{2} - \frac{i}{2}$$
___
I \/ 3
x4 = - - -----
2 2
$$x_{4} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
___
\/ 3 I
x5 = ----- - -
2 2
$$x_{5} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
___
I \/ 3
x6 = - + -----
2 2
$$x_{6} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
x6 = sqrt(3)/2 + i/2
Numerical answer
[src]
x1 = -0.866025403784439 - 0.5*i
x2 = 1.0*i
x3 = -1.0*i
x4 = 0.866025403784439 - 0.5*i
x5 = -0.866025403784439 + 0.5*i
x6 = 0.866025403784439 + 0.5*i
x6 = 0.866025403784439 + 0.5*i
The graph